CC BY
8LM-I-44
Non-instantaneous third-order polarization at low intensities
Anton Husakou1, Felipe Morales1, Maria Richter1, and Vladimir Olvo2
1 Max Born Institute, Max Born Str. 2a, 12489 Berlin, Germany 2 Department of Physics, Voronezh State University, Universitetskaya Ploshchad', 1, Voronezh, Russia, 394036
Third-order Kerr response of gases is at the core of contemporary nonlinear optics. Here we perform first-principle simulations of a hydrogen atom and demonstrate that, contrary to usual belief, the nonlinear polarization P(t) cannot be described by £0x(3)E(t)3 even at low intensities and far from resonances.
A general expression for the third-order nonlinear polarization at a given time moment t is given by (t) = £0 fffx(3(T1> T2, T3)E(.t — r1)E(t — z1—t2) E(t — z1 -t2 -T3)dT1dT2 dz3 , which can be also described in the frequency domain by J(3)(^0; <^2, w3), with m0 = m1 + m2 + m3. For pump frequencies far from resonances, instantaneous response P(t) = s0x(3E(t)3 is commonly assumed, which is equivalent to ^-independent J(3)in the frequency domain. Surprisingly, we show that this is not the case even at modest intensities with near-IR sources: not only is j(3) strongly frequency-dependent, but also there are substantial delays in the response which correspond to losses.
We perform simulations of the polarization dynamics of a hydrogen atom by numerically solving the three-dimensional time-dependent Schrodinger equation for a range of intensities. We consider 8-fs input pulse with central wavelength 800 nm, far from any resonance of the atomic hydrogen. The predicted polarization is stripped of the linear part, and third-order response is separated into fundamental-frequency and third-harmonic parts, which are described by J(3)(^0; m0, m0, —m0), and x(3)(3m0; m0, m0, m0). correspondingly. We stress that both quantities would be equal if the third-order response of the atom were instantaneous.
(TW/cm2) I (TW/cm')
Fig. 1. Nonlinear refractive index (solid curves) and delay (dashed curves) for the fundamental-frequency response (red) and third-harmonic response (green). Hydrogen atom and Yukawa potential are considered in (a) and (b), correspondingly.
One can see that the nonlinear susceptibilities J(3)(^0; m0, m0, —m0) and x(3) (3m0; m0, m0, m0) [solid curves in Fig. 1(a)] are clearly different even for very low intensities around 5 TW/cm2, where no higher-order effects can play a role. As the intensity grows the difference between the susceptibilities increases. We attribute this difference to the role of the excited states of the hydrogen atom and the interplay of the population in those states.
Additionally, one can see that at low intensities, the fundamental and third-order nonlinear responses are following carrier of the pump pulse without delay, as indicated by the dashed curves. We stress that zero delay does not mean that the response is instantaneous: the polarization carrier is not shifted from pump carrier, but its shape is still different from cos3(M01) since J(3)(^0; m0, m0, —m0) ^ j(3)(3w0; m0, m0, m0). At high intensities delays become significant; they correspond to loss which constitutes up to 30% of the nonlinear response at around 50 TW/cm2.
In order to elucidate the role of the excited states, we have repeated simulations for the Yukawa potential (with the same ionization potential), which is characterized by a significantly reduced influence of the excited states. One can see in Fig. 1(b) that the difference between j(3)(w0; m0, m0, —m0) and j(3)(3w0; m0, m0, m0) is smaller for this case, and the delays remain very close to zero.
In conclusion, we predict that the third-order response of hydrogen atoms is non-instantaneous even for low intensities and far from resonances. These findings are important both for design of the practical nonlinear optical setups and for fundamental understanding of the optical nonlinearities.
